Description Usage Arguments Details Value Note Author(s) References See Also Examples
These functions compute the probability density under some birth–death models, that is the probability of obtaining x species after a time t giving how speciation and extinction probabilities vary through time (these may be constant, or even equal to zero for extinction).
1 2 3 4 
x 
a numeric vector of species numbers (see Details). 
lambda 
a numerical value giving the probability of speciation;
can be a vector with several values for 
mu 
id. for extinction. 
t 
id. for the time(s). 
log 
a logical value specifying whether the probabilities should
be returned logtransformed; the default is 
conditional 
a logical specifying whether the probabilities
should be computed conditional under the assumption of no extinction
after time 
birth, death 
a (vectorized) function specifying how the
speciation or extinction probability changes through time (see

BIRTH, DEATH 
a (vectorized) function giving the primitive
of 
fast 
a logical value specifying whether to use faster
integration (see 
These three functions compute the probabilities to observe x
species starting from a single one after time t
(assumed to be
continuous). The first function is a shortcut for the second one with
mu = 0
and with default values for the two other arguments.
dbdTime
is for timevarying lambda
and mu
specified as R functions.
dyule
is vectorized simultaneously on its three arguments
x
, lambda
, and t
, according to R's rules of
recycling arguments. dbd
is vectorized simultaneously x
and t
(to make likelihood calculations easy), and
dbdTime
is vectorized only on x
; the other arguments are
eventually shortened with a warning if necessary.
The returned value is, logically, zero for values of x
out of
range, i.e., negative or zero for dyule
or if conditional
= TRUE
. However, it is not checked if the values of x
are
positive nonintegers and the probabilities are computed and returned.
The details on the form of the arguments birth
, death
,
BIRTH
, DEATH
, and fast
can be found in the links
below.
a numeric vector.
If you use these functions to calculate a likelihood function, it is
strongly recommended to compute the loglikelihood with, for instance
in the case of a Yule process, sum(dyule( , log = TRUE))
(see
examples).
Emmanuel Paradis
Kendall, D. G. (1948) On the generalized “birthanddeath” process. Annals of Mathematical Statistics, 19, 1–15.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28  x < 0:10
plot(x, dyule(x), type = "h", main = "Density of the Yule process")
text(7, 0.85, expression(list(lambda == 0.1, t == 1)))
y < dbd(x, 0.1, 0.05, 10)
z < dbd(x, 0.1, 0.05, 10, conditional = TRUE)
d < rbind(y, z)
colnames(d) < x
barplot(d, beside = TRUE, ylab = "Density", xlab = "Number of species",
legend = c("unconditional", "conditional on\nno extinction"),
args.legend = list(bty = "n"))
title("Density of the birthdeath process")
text(17, 0.4, expression(list(lambda == 0.1, mu == 0.05, t == 10)))
## Not run:
### generate 1000 values from a Yule process with lambda = 0.05
x < replicate(1e3, Ntip(rlineage(0.05, 0)))
### the correct way to calculate the loglikelihood...:
sum(dyule(x, 0.05, 50, log = TRUE))
### ... and the wrong way:
log(prod(dyule(x, 0.05, 50)))
### a third, less preferred, way:
sum(log(dyule(x, 0.05, 50)))
## End(Not run)

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